Algebra II Formulas Flashcards
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Questions and Answers

What is the point slope form?

  • x² + bx + c = 0
  • x³ - y³ = (x - y)(x² + xy + y²)
  • y = mx + b
  • y - y1 = m(x - x1) (correct)

What is the slope intercept form?

  • y = mx + b (correct)
  • y - y1 = m(x - x1)
  • d = √((x2 - x1)² + (y2 - y1)²)
  • x³ + y³ = (x + y)(x² - xy + y²)

Define the difference of cubes.

x³ - y³ = (x - y)(x² + xy + y²)

Define the sum of cubes.

<p>x³ + y³ = (x + y)(x² - xy + y²)</p> Signup and view all the answers

What is the quadratic formula?

<p>x = -b ± √(b² - 4ac) / 2a</p> Signup and view all the answers

What is the standard form of a quadratic equation?

<p>ax² + bx + c = 0</p> Signup and view all the answers

What is the standard form for the equation of a circle?

<p>(x - m)² + (y - n)² = r²</p> Signup and view all the answers

Define the difference of squares.

<p>x² - y² = (x + y)(x - y)</p> Signup and view all the answers

What is the representation of imaginary numbers?

<p>√(-1) = i</p> Signup and view all the answers

Define the distance formula.

<p>d = √((x2 - x1)² + (y2 - y1)²)</p> Signup and view all the answers

What is the midpoint formula?

<p>( (x1 + x2)/2 , (y1 + y2)/2 )</p> Signup and view all the answers

What condition must hold for a graph to be symmetric with respect to the x-axis?

<p>If (x,y) is on the graph, then (x,-y) is also on the graph.</p> Signup and view all the answers

What condition must hold for a graph to be symmetric with respect to the y-axis?

<p>If (x,y) is on the graph, then (-x,y) is also on the graph.</p> Signup and view all the answers

What condition must hold for a graph to be symmetric with respect to the origin?

<p>If (x,y) is on the graph, then (-x,-y) is also on the graph.</p> Signup and view all the answers

Define completing the square.

<p>x² + bx + (b/2)² = (x + b/2)²</p> Signup and view all the answers

What is the formula for the slope of a line passing through two points?

<p>m = (y2 - y1) / (x2 - x1)</p> Signup and view all the answers

What condition indicates that two lines are parallel?

<p>The slopes are equal, m1 = m2.</p> Signup and view all the answers

What condition indicates that two lines are perpendicular?

<p>The slopes are negative reciprocals, m1 = -1/m2.</p> Signup and view all the answers

Define an even function.

<p>f(-x) = f(x)</p> Signup and view all the answers

Define an odd function.

<p>f(-x) = -f(x)</p> Signup and view all the answers

What is the greatest integer function?

<p>Slanted ladder</p> Signup and view all the answers

What is a vertical shift upward in terms of a function?

<p>h(x) = f(x) + c</p> Signup and view all the answers

What is a vertical shift downward in terms of a function?

<p>h(x) = f(x) - c</p> Signup and view all the answers

What is a horizontal shift to the right in terms of a function?

<p>h(x) = f(x - c)</p> Signup and view all the answers

What is a horizontal shift to the left in terms of a function?

<p>h(x) = f(x + c)</p> Signup and view all the answers

What represents a reflection in the x-axis?

<p>h(x) = -f(x)</p> Signup and view all the answers

What represents a reflection in the y-axis?

<p>h(x) = f(-x)</p> Signup and view all the answers

How is the sum of two functions represented?

<p>(f + g)(x) = f(x) + g(x)</p> Signup and view all the answers

How is the difference of two functions represented?

<p>(f - g)(x) = f(x) - g(x)</p> Signup and view all the answers

How is the product of two functions represented?

<p>(fg)(x) = f(x)g(x)</p> Signup and view all the answers

How is the quotient of two functions represented?

<p>(f / g)(x) = f(x) / g(x)</p> Signup and view all the answers

How is the composition of two functions represented?

<p>(f o g)(x) = f(g(x))</p> Signup and view all the answers

Flashcards

Point-Slope Form

A formula used to find the equation of a line when you know a point on the line and its slope.

Slope-Intercept Form

The equation of a line where the slope and y-intercept are explicitly shown.

Difference of Cubes

A polynomial identity used to factor the difference of two perfect cubes.

Sum of Cubes

A polynomial identity used to factor the sum of two perfect cubes.

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Quadratic Formula

A formula that solves for the roots (x-values where the parabola crosses the x-axis) of a quadratic equation.

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Quadratic Equation

A second-degree polynomial equation written in the form ax^2 + bx + c = 0.

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Standard Form of a Circle

The standard form of a circle's equation, showing its center and radius.

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Difference of Squares

A factoring identity used to factor the difference of two perfect squares.

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Imaginary Numbers

A type of number defined as the square root of negative one, represented by the symbol 'i'.

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Sum of Squares

A mathematical expression involving the squares of numbers, but it cannot be factored using real numbers.

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Distance Formula

Used to find the distance between two points in a coordinate plane.

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Midpoint Formula

A formula used to find the midpoint between two points in a coordinate plane.

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X-axis Symmetry

A property of a graph where it is mirrored about the x-axis.

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Y-axis Symmetry

A property of a graph where it is mirrored about the y-axis.

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Origin Symmetry

A property of a graph where it is mirrored about the origin (0,0).

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Completing the Square

This method transforms a quadratic equation into its vertex form by manipulating the expression to create a perfect square trinomial.

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Principal Square Root

The positive square root of a number when it is non-negative. For negative numbers, it is expressed in terms of an imaginary unit.

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Slope of a Line

The measure of the steepness of a straight line, representing its rate of change.

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Parallel Lines

When two lines have the same slope, they are always the same distance apart and never intersect.

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Perpendicular Lines

When the slopes of two lines are negative reciprocals of each other, they intersect at a right angle.

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Even Function

A function where the output is the same for both positive and negative values of the input.

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Odd Function

A function where the output for a negative input is the negative of the output for the positive input.

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Greatest Integer Function

A function that gives the greatest integer less than or equal to the input.

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Vertical Shifts

A shift of a function's graph vertically; it moves the graph up or down.

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Horizontal Shifts

A shift of a function's graph horizontally; it moves the graph left or right.

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Reflections

A transformation that flips the graph of a function across an axis.

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Composition of Functions

Combining two functions by applying one function to the output of the other.

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Operations on Functions

Performing arithmetic operations like addition, subtraction, multiplication, and division on functions.

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Study Notes

Algebra II Formulas

  • Point Slope Form:
    Useful for writing the equation of a line given a point ((x_1, y_1)) and the slope (m). Formula: (y - y_1 = m(x - x_1)).

  • Slope Intercept Form:
    Represents linear equations, where (m) is the slope and (b) the y-intercept. Formula: (y = mx + b).

  • Difference of Cubes:
    A polynomial identity used to factor the difference of two cubes. Formula: (x^3 - y^3 = (x - y)(x^2 + xy + y^2)).

  • Sum of Cubes:
    A polynomial identity for factoring the sum of two cubes. Formula: (x^3 + y^3 = (x + y)(x^2 - xy + y^2)).

  • Quadratic Formula:
    Used to find the roots of a quadratic equation (ax^2 + bx + c = 0). Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).

  • Quadratic Equation:
    A second-degree polynomial equation expressed as (ax^2 + bx + c = 0).

  • Standard Form for Equation of a Circle:
    Represents a circle's equation, where ((m,n)) is the center and (r) is the radius. Formula: ((x - m)^2 + (y - n)^2 = r^2).

  • Difference of Squares:
    A factoring identity for the difference of two squares. Formula: (x^2 - y^2 = (x + y)(x - y)).

  • Imaginary Numbers:
    Defined by the square root of negative one, represented as (i), where (i = \sqrt{-1}).

  • Sum of Squares:
    Representation involving squares of numbers, though it cannot be factored into real numbers. Formula: (x^2 + y^2 = (x^2 + xy + y^2)).

  • Distance Formula:
    Used to calculate the distance (d) between two points ((x_1, y_1)) and ((x_2, y_2)). Formula: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}).

  • Midpoint Formula:
    Determines the midpoint between two points. Formula: (\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)).

  • Symmetry in Graphs:

    • X-axis: A graph is symmetric with respect to the x-axis when ((x, y)) corresponds to ((x, -y)).
    • Y-axis: Symmetry around the y-axis occurs when ((x, y)) corresponds to ((-x, y)).
    • Origin: Symmetry about the origin is present when ((x, y)) corresponds to ((-x, -y)).

  • Completing the Square:
    A method to transform quadratic equations into vertex form. Formula: (x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2).

  • Principal Square Root of a Number:
    For a negative number (-a), the square root is defined in terms of imaginary numbers. Formula: (\sqrt{-a} = \sqrt{ai}).

  • Slope of a Line:
    The slope (m) of a line passing through two points can be calculated. Formula: (m = \frac{y_2 - y_1}{x_2 - x_1}).

  • Parallel Lines:
    Two lines are parallel if their slopes are equal, represented as (m_1 = m_2).

  • Perpendicular Lines:
    Two lines are perpendicular if their slopes are negative reciprocals, represented as (m_1 = -\frac{1}{m_2}).

  • Even Function:
    A function (f) is even if it satisfies (f(-x) = f(x)).

  • Odd Function:
    A function (f) is odd if it satisfies (f(-x) = -f(x)).

  • Greatest Integer Function:
    Often referred to as the "floor function," visualized like a slanted ladder.

  • Vertical Shifts:

    • Upward Shift: (h(x) = f(x) + c).
    • Downward Shift: (h(x) = f(x) - c).

  • Horizontal Shifts:

    • Right Shift: (h(x) = f(x - c)).
    • Left Shift: (h(x) = f(x + c)).

  • Reflections:

    • In the x-axis: (h(x) = -f(x)).
    • In the y-axis: (h(x) = f(-x)).

  • Operations on Functions:

    • Sum: ((f + g)(x) = f(x) + g(x)).
    • Difference: ((f - g)(x) = f(x) - g(x)).
    • Product: ((fg)(x) = f(x)g(x)).
    • Quotient: ((f / g)(x) = \frac{f(x)}{g(x)}).

  • Composition of Functions:
    Describes the output when a function (g) is applied to the input of another function (f). Formula: ((f \circ g)(x) = f(g(x))).

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Test your knowledge of key Algebra II formulas with this set of flashcards. Each card presents a crucial mathematical concept, such as point-slope form and the quadratic formula, making it a perfect tool for review and practice. Strengthen your understanding of algebraic principles effectively!

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